Determine a Matriz Adjunta das seguintes matrizes:
\left[\begin{array}{lll}1 & 0 & 0 \\\\ 2 & 1 & 0 \\\\ 3 & 4 & 1\end{array}\right]

Passo 1

O enunciado nos dá a matriz
A = \begin{pmatrix} 1 & 0 & 0\\\\ 2 & 1 & 0\\\\ 3 & 4 & 1 \end{pmatrix}
e pede a matriz adjunta de A. Para isso precisamos usar a definição: primeiro calculamos a matriz de cofatores C, cujas entradas são
C_{ij} = (-1)^{i+j}\det(M_{ij}),
onde M_{ij} é o menor obtido eliminando-se a linha i e a coluna j de A. Depois a matriz adjunta é
\operatorname{adj}(A) = C^{T}.

Passo 2

Vamos calcular cada cofator. Eliminando a linha 1 e a coluna 1, 2 e 3, temos:
\begin{aligned} M_{11}&=\begin{vmatrix}1&0\\\ 4&1\end{vmatrix}=1\cdot1-0\cdot4=1, \quad&C_{11}&=(+1)\cdot1=1,\\\ M_{12}&=\begin{vmatrix}2&0\\\ 3&1\end{vmatrix}=2\cdot1-0\cdot3=2, \quad&C_{12}&=(-1)\cdot2=-2,\\\ M_{13}&=\begin{vmatrix}2&1\\\ 3&4\end{vmatrix}=2\cdot4-1\cdot3=5, \quad&C_{13}&=(+1)\cdot5=5. \end{aligned}
Eliminando a linha 2 e as colunas:
\begin{aligned} M_{21}&=\begin{vmatrix}0&0\\\ 4&1\end{vmatrix}=0\cdot1-0\cdot4=0, \quad&C_{21}&=(-1)\cdot0=0,\\\ M_{22}&=\begin{vmatrix}1&0\\\ 3&1\end{vmatrix}=1\cdot1-0\cdot3=1, \quad&C_{22}&=(+1)\cdot1=1,\\\ M_{23}&=\begin{vmatrix}1&0\\\ 3&4\end{vmatrix}=1\cdot4-0\cdot3=4, \quad&C_{23}&=(-1)\cdot4=-4. \end{aligned}
Eliminando a linha 3 e as colunas:
\begin{aligned} M_{31}&=\begin{vmatrix}0&0\\\ 1&0\end{vmatrix}=0\cdot0-0\cdot1=0, \quad&C_{31}&=(+1)\cdot0=0,\\\ M_{32}&=\begin{vmatrix}1&0\\\ 2&0\end{vmatrix}=1\cdot0-0\cdot2=0, \quad&C_{32}&=(-1)\cdot0=0,\\\ M_{33}&=\begin{vmatrix}1&0\\\ 2&1\end{vmatrix}=1\cdot1-0\cdot2=1, \quad&C_{33}&=(+1)\cdot1=1. \end{aligned}
Logo a matriz de cofatores é
C = \begin{pmatrix} 1 & -2 & 5\\\\ 0 & 1 & -4\\\\ 0 & 0 & 1 \end{pmatrix}.

Passo 3

A adjunta é a transposta de C:
\operatorname{adj}(A)=C^{T}= \begin{pmatrix} 1 & 0 & 0\\\\ -2 & 1 & 0\\\\ 5 & -4 & 1 \end{pmatrix}.

Resposta

\operatorname{adj}\bigl(\begin{pmatrix} 1 & 0 & 0\\\\ 2 & 1 & 0\\\\ 3 & 4 & 1 \end{pmatrix}\bigr) = \begin{pmatrix} 1 & 0 & 0\\\\ -2 & 1 & 0\\\\ 5 & -4 & 1 \end{pmatrix}.

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